MATH THIRD QUARTER
Use inductive reasoning to determine the next number in the sequence. 1, 2, 4, 8, 16, __
32
What is the value of x? (triangle)
40
In the figure below, AD__CD.
>
In the given figure below, what theorem/definition/postulate should be used as a reason for ∠ACB≅∠DCE?
Definition of Vertical Angles
One may say that in the figure below, m∠WUV+m∠UVW=m∠EWV because of the --------_.
Exterior Angle Theorem
What axiom states the following: The whole equals the sum of its parts.
Partition
Which of the following directly proves that 1/a>0 if a>0?
The reciprocal of any positive number is also a positive number The quotient of 1 and any positive number is also a positive number
What is an angle bisector?
an line or part that divides an angles into two equal parts
Two triangles are said to be congruent if ...
their corresponding sides and corresponding angles are congruent
Given: ΔABC ≅ ΔXYZ ∠CAB= 10x + 25 ∠ZXY= 5x + 40 What is the measure of ∠CAB?
55
ΔMAT ≅ ΔHEN. If MT = 5 cm, EN = 6 cm. What is the measure of TA?
6 cm
In the figure below, what is the measure of ∠x?
69
A possible length for the third side of a triangle with sides 6 and 12 would be
8
AB bisects ∠DAC. If ∠DAB=40, what is the measure of ∠DAC
80
Which of the following proves the statement below using direct proof? The product of two odd numbers is an odd number. (Note: Let nn be any integer.)
(2n + 1)(2n + 1) = 4n^2 + 4n + 1 (2n + 1)(2n + 1) = 2(2n^2 + 2n) + 1
What is an acceptable length for the third side of a triangle with sides 3 and 4?
5
Given: ΔABC≅ΔXYZ AB = x+4 BC=2x+2 AC=3x+3 XZ=x+5 How long are XY, YZ, and XZ
5 cm, 4 cm, and 6 cm
Given: ΔMAT ≅ ΔHEN. If ∠MAT = 40°, ∠MTA = 60°, and ∠TMA = 80°, what is the measure of ∠EHN
80
What are perpendicular lines?
A pair of intersecting lines forming right angles at the point of intersection
Using the given below, determine which additional pair of sides should be congruent so that the two triangles are congruent by SAS congruence postulate.
AB and DF
Complete the proof below: Given: AD ≅ DC ∠ADB≅∠CDB Prove: AB ≅ BC
AB ≅ BC 4 Given 1 SAS congruence postulate 3 BD ≅ BD 2
If ΔABC ≅ ΔDEFΔ, which is congruent to DF?
AC
Which triangle congruence postulate is referred to by the statement "If two angles and an included side of one triangle are congruent to the corresponding two angles and included side of another triangle, then the triangles are congruent."?
ASA congruence postulate
Explain why the following statement is a result of an induction. Roy is a football player. All football players weigh 150 pounds. Therefore, Roy weighs 150 pounds.
It is a result of induction because you begin with some data, and then determine the conclusion can be derived from those data
AB is perpendicular to CD
CD bisects AB
m<1 > m<2
Corollary of the Hinge Theorem
What does CPCTC stand for?
Corresponding Parts of Congruent Triangles are Congruent
Which of the following statements is true about the figure below?
DEF is larger than each of FGE and EFG
Suppose that the following figure exhibits the Hinge Theorem, What necessary condition must be satisfied in the figure below for us to conclude that m∠A<m∠Dm?
DF must be 11
Which is the longest side of ΔDEV?
DV
Dave rides 3 miles north then 1.5 miles N120°W. Ellen on the other hand rides 3 miles south then 1.5 miles east. Who is farther from the park?
Dave
What kind of reasoning do you use when you determine what else would have to be true before making a conclusion?
Deductive reasoning
BM = CM
Definition of Median
If ABCD is a square, what theorem/definition/postulate should be used as a reason for ∠BAC≅∠BDC
Definition of Right Angles
BDA = 180 - 110
Definition of Supplementary Angles
Given: AD=CD, m∠BDC = 110 Prove: CB > AD 3. ∠BCD and ∠BDA are supplementary angles
Definition of Supplementary Angles
Identify whether the following statement is true or false, and explain why. If BE is perpendicular to AC at D and DF bisects ∠ABD, then ∠CDF=45
False; CDF = 135
CB > AD
Hinge Theorem
Which of the following is the correct description of "construction" in geometry?
It is done by the use of a straight-edge and a compass
Write an indirect proof. If 3x > 12, then x > 4
Let x ≤ 4
Which of the following uses deductive reasoning to prove that 4 is a factor of 24?
Listing the multiples of 4
A form used in proving a statement using direct proof.
Modus Ponens
What axiom states the following: If equal quantities are multiplied by equal quantities, the products are equal. That is, if a = b and c = d, then ac = bd
Multiplication
Which of the following statements is not true?
No geometric figure can be moved without change in size or shape.
Which of the following is true about perpendicular lines?
Perpendicular lines form right angles
It refers to a sequence of statements to make a conclusion.
Proof
Which of the following is an indirect proof?
Proof by Contraposition Proof by Contradiction
Draw out the correct conclusion from the following statements: 1. Points M, J, and B lie on the same line and MJ = JB. 2, R is the midpoint of MB
R and J represent the same point
It refers to the process of drawing conclusions from facts.
Reasoning
What are the longest and shortest side of the triangle below?
The longest side is opposite the 115 angle; the shortest sides is opposite is opposite the 25
Which of the following does NOT describe an isosceles triangle?
The measure of the congruent sides does not exceed 10
These are statements proven by axioms.
Theorems
Consider the following axiom set: Axiom 1: Every cat has at least three cans of catnips. Axiom 2: Every can of catnips has at least two cats. Axiom 3: There exist at least two cats. Choose the correct proof for the axiom, "There exist at least two cans of catnips."
There exist at least two cats. Every cat has at least three cans of catnips; thus, there exist at least two cans of catnips
What axiom states the following: Things equal to the same or equal things are equal to each other. That is, if a = b and c = b, then a = c.
Transitive
Identify whether the following statement is true or false, and explain why. The bisector of an obtuse angle forms two congruent acute angles.
True; an obtuse angle measures less than 180, therefore when divided into two equal parts, the resulting angles will be acute angles.
Identify whether the following statement is true or false, and explain why. If ∠ABC=60∘, then ∠EBF=60∘
True; if ABC = 60, then 1 = 2 = 30. Since BC bisects EBF and 2 = 30, therefore 3 = 30. Therefore EBF = 60
Which of the following can only be described to have meanings.
Undefined terms
Which of the following is FALSE?
Vertical angles are acute angles.
Use deductive reasoning to determine which statement best fits the following scenario. All humans are mortal. I am human. Therefore, --------_.
I am mortal
It takes me an hour to get to the mall. If I leave at 5 o'clock, then --------_.
I will reach the mall by 6 o'clock
Which of the following forms applies in deductive reasoning?
If A = B and B = C, then A = C.
Which of the following is a correct statement regarding the figures below?
If BC and DE are congruent, then the two triangles are congruent by SAS postulate
What do you call the type of reasoning wherein you draw a conclusion from observations or facts?
Inductive reasoning
Explain why the following statement is a result of an induction. All chickens that we have seen are white, therefore all chickens are white.
It is a result of induction because you begin with some data, and then determine the conclusion can be derived from those data
Explain why the following statement is a result of an induction. Jake just moved to Canada. He has red hair. Therefore people from Canada have red hair.
It is a result of induction because you begin with some data, and then determine the conclusion can be derived from those data
An exterior angle is always --------_ to one of the interior angles of a triangle.
adjacent
In writing a proof regarding statements on triangle congruence, which of the following is written first?
all the given
What are the undefined terms in the following axiom? Every bee belongs to two and only two hives.
bee, hives, belongs
What is the relation in the following axiom? Every path is between two ants.
between
What are the undefined terms in the following axiom? There are at least two buildings on campus.
buildings, campus, on
Consider the following axiom set: Axiom 1: Every cat has at least three cans of catnips. Axiom 2: Every can of catnips has at least two cats. Axiom 3: There exist at least two cats. What are the undefined terms in the axiom set?
cat, can of catnips, has
This is the relation from a set of given statements.
conclusion
It is a way of showing the truth or falsehood of a given statement. This proof usually uses facts and axioms.
direct proof
It refers to some statements that support another statement.
premise
Given a triangle with lengths measuring 9 and 12, the third side would be of length 11.5. Is this true?
sometimes
In stating triangle congruence, which of the following should be considered?
the order of the letters in the statement should present the corresponding congruent parts of the triangles in the same order
Choose the statement that completes the following sentence: If two polygons of equal area both quadruple in size, then ____
the two polygons will still have the same area
Look at the figures below. Use inductive reasoning to determine which should be the next figure.
vertical line image
ΔMAT ≅ ΔHEN. If MT = (5x - 4) cm, HN = (3x + 6) cm, what is the value of x?
x = 5
In a triangle, one side has length x, and a side adjacent to it has twice that length. What would be the range of possible values for the third side y in terms of x?
x<y<3x
Prove that if xx and yy are even, then x2+y2x2+y2 is divisible by 4.
x^2 + y^2 = 4(m^2 + n^2)
Which additional pair of parts (angle or side) should be congruent in the triangles above so that the triangles are congruent by ASA congruence postulate?
∠A and ∠F
If ΔABC ≅ ΔXYZ, which of the following is NOT correct?
∠A is congruent to ∠Z
What conclusion follows when the transitive axiom is applied to the following data? 1. ∠A, ∠B, and ∠C measure 90° each 2. A right angle is an angle that measures 90°.
∠A, ∠B, and ∠C are right angles
If AB bisects ∠DAC, name the two congruent angles formed
∠DAB and ∠CAB
Determine the next number in the sequence 2, 4, 6, 8, 10, ....
12
In the figure below, what is the measure of ∠x?
12
Which of the following sets of lengths could form a triangle?
15,12,9
What may be concluded about ∠BUG and ∠PET?
m∠BUG > m∠PET
AB > AC
given
Prove that id 2r + 3 is not equal to 17, then r is not equal to 7.
But 2r + 3 is not equal to 17 (5th line)
Arrange the following steps to show the how to prove statements on triangle congruence.
1. Write the given information as the first statement(s) .2. Derive other necessary information from the given, diagram, and/or problem statement .3. Once the corresponding congruent parts are enough, write the triangle congruence statement together with the supporting triangle congruence postulate. 4. You may now prove that the corresponding parts asked are congruent by CPCTC.
Find the value of x in the figure below.
11
What is the measure of the exterior angle adjacent to ∠WUV?
113
Choose the statement that completes the following statement: If one straight angle measures 180° and another straight angle measures π rad, then we know that
180° = π
The following lengths can be used for the third side of a triangle with lengths 4 and 7 except
2
Fill in the blank with a possible value. If n^2 is even, then n^2 is divisible by
2, 4
All numbers ending in 0 or 5 are divisible by 5. The number 20 ends with 0. Therefore, --------_.
20 is divisible by 5
Given: ΔABC ≅ ΔXYZ AB=5x+2 cm BC=4x-15 cm XY=3x+20 How long is BC?
21 cm
Determine the next probable number in the list using inductive reasoning. 3, 7, 11, 15, 19, 23, __
27
What would be the sum of the measures of ∠EWV, ∠WVU, and WUV in the figure below?
288
Which of the following completely proves that for any integers r and s, 2r−6+10s2r−6+10s is even?
2r - 6 + 10s = 2(r-3 +5s)
Consider the following axiom set: Axiom 1: Every cat has at least three cans of catnips. Axiom 2: Every can of catnips has at least two cats. Axiom 3: There exist at least two cats. What is the minimum number of cans of catnips?
3
In ΔRST, 9RS=3x−9, ST=5x+2, and RT=9x−17. What is the range of possible value of x?
4 < x < 10
What is the measure of ∠U in the figure below?
40
Which of the following statements use deductive reasoning to prove that ∠A is an acute angle?
All acute angles are less than 90 degrees. ∠A is less than 90 degrees. Therefore, ∠A is an acute angle
Be careful around bees, they might sting you. Then, __.
All bees might sting
By indirect proof, which of the following must be done to prove that if 2x−3>7, then x>5?
Assume the negation of x > 5. If the negation of 2x - 3 > 7 is true, then the original statement is true. Assume the negation of x > 5. If the statement contradicts 2x - 3 > 7, then the original statement is true.
These are statements accepted as true and do not need a proof.
Axioms
Which conclusion may be made from the figure below?
BC < CD < DE
If the two triangles below are congruent by SAS congruent postulate, which of the following is true?
BC ≅ ED
Select the conclusions that follow when the partition axiom is applied to the following data:
BD = 6 + 7 = 13 AD = 4 + 6 + 7 = 17 AC = 4 + 6 = 10
Using the distributive property of multiplication, (x+2y)2=x2+4xy+4y2(x+2y)2=x2+4xy+4y2. Why is this a deductive reasoning?
Because it states facts and other premises (i.e the solution) that support that the statement is true
AM = AM
Reflexive Property
Complete the table by selecting the appropriate reason for the given statement. Given: AD=CD, m∠BDC = 110 Prove: CB > AD 2. BD = BD
Reflexive Property
In the given figure below, what theorem/definition/postulate should be used as a reason for CD ≅ CD
Reflexive property
Complete the two-column proof. AB is perpendicular to CD CB ≅ BD Prove: ∠C ≅ ∠D
Reflexive property 2 SAS congruence postulate 3 Angles formed by perpendicular lines are right angles. 1 CPCTC 4
If three sides of a triangle are congruent to the corresponding three sides of another triangle, then the triangles are congruent by what supporting postulate?
SSS congruence postulate
Which triangle congruence postulate can be used to prove that the two triangles are congruent?
SSS congruence postulate
BDA = 70
Simplification
Is nm - m composite for n > 2 and m > 1
Yes, since nm - m is factorable into m (n - 1) without m or n - 1 being equal to 1
Are the two triangles congruent? Why or Why not?
Yes, there are two parts of congruent angles and the BD of ΔABD is congruent to side BD of ΔCBD
In ΔRST, RS=12, and ST=8. What is the least possible value of the third side RT?
a little longer than 4.
In ΔRST, RS=12, and ST=8. What is the greatest possible value of the third side RT?
a little shorter than 20
If a + b > 100, use the indirect proof to show that either a > 50 or b > 50.
a ≤ 50 and b ≤ 50
Using direct proof, which of the following shows that the sum of two rational numbers is a rational number?
a/b + c.d = ad + bc / bd; for any integers a,b,c,d and b,d are not equal to 0.
Which makes the following statement true? If a>0, b>0, and a>b, then -----_ .
a/b > 1
Complete the table by selecting the appropriate reason for the given statement. Given: AD=CD, m∠BDC = 110 Prove: CB > AD
given
The measure of an exterior angle is --------_ any of its remote interior angles.
greater than
In the figure below, m∠1---m∠2
gt
Which of the following statements is true about the figure? (diamond)
he triangles are congruent by SSS postulate.
What is/are the element(s) in the following axiom? For any herd, there exists a sheep not in the herd.
herd, sheep
An unproven statement that might be true.
hypothesis
Which of the following is/are correct?
if two intersecting lines form a 90 angle, then the lines are perpendicular If a ray bisects a right angle, the measure of the congruent angles formed in 45 An angle can only have one angle bisector
A kind of proof used by proving a statement in which the conclusion is assumed as false, and the assumption leads to an impossibility which makes the original statement true.
indirect proof
In the figure below, what can you conclude regarding AB and AC?
insufficient information
In indirect proof, the contrapositive always assumes the ______of the conclusion.
negation
Given: a+b is odd. Prove: a or b is odd. By indirect proof, which of the following is the correct initial assumption?
neither a nor b is odd
Would the set of numbers 11, 9, and 20 form the sides of a triangle?
never
How many straight lines can be drawn through any two points?
only one
